Rats are probably a long term problem in Tehran, but reports of their size are probably exaggerated.
A news source closer to Iran is Dubai based The National which reports
Ismail Kahram, an environmental adviser to the city council ... said cats are now smaller than some of Tehran's rats, which can weigh up to 5kg.
In some versions of this story and in some unrelated reports, Ismail Kahram is described as a professor at Tehran Azad University.
Wikipedia isn't a great source but claims of Rattus Norvegicus (brown rat)
Adult body weight averages 550 g (19 oz) in males and about 350 g (12 oz) in females, but a very large individual can reach 900 g (32 oz). Rats weighing over 1 kg (2.2 lb) are exceptional, and stories of rats as big as cats are exaggerations
and of Rattus Rattus (black rat)
A typical adult black rat is 12.75–18.25 in (32.4–46.4 cm) long, including a 6.5–10 in (17–25 cm) tail, and weighs 4–12 oz (110–340 g)
University of Michigan Museum of Zoology says
of Rattus Norvegicus (brown rats)
On average, these rats reach nearly 400 mm nose-to-tail, and weigh 140 to 500 g. Males are usually larger than females.
Brown rats (R. norvegicus) raised in captivity can reach 800g but it seems unlikely that large numbers of wild rats do so.
In Investigation of Contamination of Wild Rats (Rattus rattus) from Tehran City to Antibiotic Resistant Enterobacteriaceae in 2009 says
Background and purpose: Wild rats are considered as an important spreading factor of contamination in urban areas. During recent years, the wild rats were extensively distributed in Tehran
Materials and methods: Forty wild rats were trapped in the North, South, East, West and Central areas of Tehran and transferred to the laboratory alive.
So the species prevalent in Tehran is Rattus Rattus - the black rat, which is normally significantly smaller than the brown rat.
A Google translation of the Persian language report doesn't reveal any suggestion that the rats were over ten times bigger than "normal" specimens in 2009.
The origin of the story is unclear. It seems a bit circular.